Combined Trajectory, Propulsion, and Battery Mass Optimization for
Solar-Regenerative High-Altitude Long Endurance Unmanned
AircraftAll Faculty Publications
Kevin R. Moore Brigham Young University,
[email protected]
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Original Publication Citation Gates, N. S., Moore, K. R., Ning, A.,
and Hedengren, J. D., “Combined Trajectory, Propulsion, and Battery
Mass Optimization for Solar-Regenerative High-Altitude Long
Endurance Unmanned Aircraft,” AIAA Scitech 2019 Forum, San Diego,
CA, Jan. 2019. doi:10.2514/6.2019-1221
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BYU ScholarsArchive Citation Gates, Nathaniel S.; Moore, Kevin R.;
Ning, Andrew; and Hedengren, John D., "Combined Trajectory,
Propulsion, and Battery Mass Optimization for Solar-Regenerative
High-Altitude Long Endurance Unmanned Aircraft" (2019). All Faculty
Publications. 2982.
https://scholarsarchive.byu.edu/facpub/2982
This conference paper is available at BYU ScholarsArchive:
https://scholarsarchive.byu.edu/facpub/2982
Optimization for Solar-Regenerative High-Altitude
Long Endurance Unmanned Aircraft
Nathaniel S. Gates∗, Kevin R. Moore†, Andrew Ning‡, and John D.
Hedengren§
Brigham Young University, Provo, UT, 84602, USA
Combined optimization of propulsion system design, flight
trajectory planning and bat- tery mass optimization for
solar-regenerative high-altitude long endurance (SRHALE) air- craft
through a sequential iterative approach yields an increase of 20.2%
in the end-of-day energy available on the winter solstice at 35°N
latitude, resulting in an increase in flight time of 2.36 hours.
The optimized flight path is obtained by using nonlinear model pre-
dictive control to solve flight and energy system dynamics over a
24 hour period with a 15 second time resolution. The optimization
objective is to maximize the total energy in the system while
flying a station-keeping mission, staying within a 3 km radius and
above 60,000 ft. The propulsion system design optimization
minimizes the total energy required to fly the optimal path. It
uses a combination of blade element momentum theory, blade
composite structures, empirical motor and motor controller mass
data, as well as a first order motor performance model. The battery
optimization seeks to optimally size the battery for a circular
orbit. Fixed point iteration between these optimization frameworks
yields a flight path and propulsion system that slightly decreases
solar capture, but signif- icantly decreases power expended. Fully
coupling the trajectory and design optimizations with this level of
accuracy is infeasible with current computing resources. These
efforts show the benefits of combining design and trajectory
optimization to enable the feasibility of SRHALE flight.
Nomenclature
N Wing surface normal ¯SN Sun direction vector γ Flight path angle
(rad) µsolar Obliquity factor φ Bank angle (rad) φs Solar azimuth
(rad) ψ Heading Angle (rad) ρ Air density (kg/m3) θ Pitch angle
(rad) θs Solar zenith (rad) AR Aspect ratio CD Drag coefficient CL
Lift coefficient D Drag (N) Dprop Propeller diameter (m)
Ebattery Energy stored in battery (kWh) einv Inviscid span
efficiency Epotential Potential energy storage (kWh) g Gravity
(m/s2) h Altitude (m) I0 Motor no load current (amps) Kv Motor
rotational constant (rad/s) L Lift (N) m Aircraft mass (kg) mmotor
Motor mass (kg) Pbattery Power to battery (W ) Ppayload Power
required for payload (W ) Ptotal Total power balance (W ) q Dynamic
pressure (Pa) r Propeller radial position (m)
∗Graduate Student, Department of Chemical Engineering, 330 EB,
Provo, UT 84602, AIAA Student Member †Graduate Student, Department
of Mechanical Engineering, 350 EB, Provo, UT 84602, AIAA Student
Member ‡Assistant Professor, Department of Mechanical Engineering,
350 EB, Provo, UT 84602, AIAA Senior Member §Associate Professor,
Department of Chemical Engineering, 330 EB, Provo, UT 84602
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American Institute of Aeronautics and Astronautics
R0 Motor no load resistance (ohms) S Wing surface area (m2) T
Thrust (N)
V Velocity (m/s) x Position North (m) y Position East (m)
I. Introduction
Satellites and manned aircraft are currently heavily used resources
in areas such as public communication, surveillance, agriculture,1
weather, and environmental monitoring. Although effective, they
come with their share of concerns. One of the primary issues is
cost. Both satellites and manned aircraft systems cost millions of
dollars to manufacture and maintain. There are also environmental
concerns that accompany the fossil- fuel-powered propulsion systems
and satellites’ contributions to space debris in orbit.2 Solar
Regenerative High Altitude Long Endurance (SRHALE) aircraft have
the potential to provide a more cost effective, energy efficient,
and environmentally friendly alternative to space satellites.
Using aircraft as pseudo-satellites has been considered a feasible
possibility only recently. However, the technology leading up to
this consideration has long been in development. Electric, battery
powered aircraft have been experimented with since the 1800s, and
the addition of solar panels to electric aircraft has been studied
since the 1970s.3,4
Experimentation with solar-electric aircraft has significantly
progressed since the 1800s. One notable SRHALE aircraft is NASA’s
Helios Prototype, a remotely piloted flying wing, which achieved a
world record altitude of 96,843 feet in 2001, but experienced
catastrophic failure over the Pacific Ocean in 2003.5 Other recent,
notable SRHALE aircraft ventures include Qinetiq’s Zephyr (which
holds the current heavier than air aircraft endurance record),6
Facebook’s Aquila,,7 Aurora’s Odysseus,8 and Boeing’s
SolarEagle.9
Topics relevant to SRHALE optimization and design are present in
the published literature, and show many unique ways to increase the
feasibility of SRHALE flight. These include structures, controls,
multi- disciplinary design optimization, and path
optimization.
Unique structural considerations have been found to be of
importance for HALE aircraft. McDonnell, Cesnik, Palacios, and
Reichenbach review conventional structural design procedures and
their application to highly flexible aircraft generally proposed
for HALE applications.10–12 Hesse and Palacios also propose
reduced-order aeroelastic models for flexible aircraft.13
Special considerations in HALE aircraft controls have also been
studied. Kubica and Livet developed flight control laws for
flexible aircraft. Furthermore, Caverly et. al. as well as
Haghighat, Liu, and Martins present methods for gust load
alleviation for flexible HALE aircraft1415 The questions of
inherent feasibility and multidisciplinary interactions of SRHALE
aircraft have also been considered. Brandt and Gilliam exhibit a
basic method for the analysis of solar powered aircraft.16 Baldock
and Mokhtarzadeh-Dehghan presented a methodology for determining
the feasibility of SRHALE aircraft and suggested the use of a
braced-wing concept.17 Morrisey performed Multidisciplinary Design
Optimization (MDO) to investigate a pinned wing SRHALE concept.18
Ozoroski, Nickol, and Guynn have created an analysis tool for
station keeping considerations for both SRHALE airplanes and
airships.19
A possible, though comparatively less-explored region of SRHALE
technology is in trajectory optimiza- tion. Martin showed that, in
comparison to cases without trajectory optimization, solar aircraft
can increase their energy capture and efficiency of energy use.20
To make use of this idea, in this paper the aircraft path is
optimized dynamically at a fine time resolution, balancing
competing objectives to fly efficiently and increase solar energy
capture in order to maximizing the total energy in the system. It
also stores potential energy by climbing when the battery is full.
The net result is to increase the state of charge at the end of the
day.
Despite the presence of quality research and development in SRHALE
technology, it has been found that there is room for increased
detail in a highly multi-disciplinary, system-level design
optimization. Published research in the areas of SRHALE aircraft
have generally included a small subset of the many models that go
into a complete aircraft. Furthermore, most tools developed thus
far have been primarily for analysis, and have not been applied to
optimization-based design. Some of the published research has been
found to be more complete in the inclusion of a large number of
important models, as well as optimization considerations, but none
capture the design space between dynamics and design as proposed
here. In this research, a unique contribution to the already
present literature is made by combining trajectory and propulsion
design for SRHALE aircraft.
The importance of this path and design optimization is highlighted
by the computational expense of
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American Institute of Aeronautics and Astronautics
combining the two. A full day of dynamic optimization with a
simplified aircraft model at a 15 second time step requires 5,760
successful horizon optimizations. Each horizon requires on average
25 to 100 iterations to converge, and each iteration makes multiple
function calls to evaluate the states. If the full aircraft model
(including structures) and mid-fidelity aerodynamics (including
flexible wing stability constraints) were to be implemented with a
5 second function evaluation time, the total optimization would
take 800 hours (33.3 days). This makes the unlikely assumptions
that the optimization problem converges in the same amount of
iterations and includes analytic derivatives. Parallelization is
extremely limited due the inherent sequential progression of a
dynamic system, and would only be able to affect the time of each
individual iteration. Efforts to parallelize these iterations have
failed to meaningfully improve the convergence speeds, with the
slight reduction in an already fast iteration time failing to
offset the expense of increased overhead.
In this paper, it is shown how an extremely simple point-based
iteration method can significantly increase the feasibility of an
SRHALE aircraft design at a fraction of the computational cost. By
extracting the simplified (or condensed) physics of the propulsion
system in the form of continuous splines, the simple dynamic
aircraft performance and propulsion models can be evaluated several
orders of magnitude faster, which shrinks the total optimization
time-frame from months to hours. First, the models will be
described, along with the assumptions and limitations that
accompany them. These include models of aircraft dynamics, aircraft
drag and lift, motor efficiency, propeller efficiency, and solar
panel efficiency. Next, the optimization problem and setup for the
combined iterative approach will be explained, which is comprised
an iterative approach between the three subsystems of propulsion,
path, and battery mass optimization. Finally, the results and
conclusions will be presented, which show significant increases in
HALE mission performance.
II. Model Description
The models used in this work were chosen in order to achieve
sufficient accuracy while remaining compu- tationally tractable.
The design of the aircraft was kept largely constant, save for
changes to the propulsion system and battery mass. Aircraft
structural constraints were assumed to be satisfied despite slight
changes in the overall aircraft mass. In addition, this study was
performed in the absence of wind.
A. Aircraft Dynamic Model
The aircraft dynamic model is used in the path optimization, which
solves a nonlinear programming problem with inequality constraints.
It describes the fundamental physics of the aircraft flight, and
must be solved simultaneously across a large time horizon.
1. Aircraft Parameters and Mission Conditions
The aircraft definition follows the same approach taken in our
previous work,20 where the SRHALE unmanned aircraft is modeled as a
large flying wing similar to Facebook’s Aquila. The parameters are
reproduced here in Table 1.
Similarly, the flight mission conditions are specified to mimic an
SRHALE aircraft acting as a fixed node in a communications network.
The flight altitude is constrained to be above U.S. regulated
airspace (18,288 m, or 60,000 ft), and the flight radius to be
within 3 km from a central point.20 The time and location are
chosen to represent a particularly demanding case within the range
of applicable SRHALE aircraft missions: the 2017 winter solstice
(December 21), the darkest night of the year, and 35°N latitude,
below which a large portion of the world’s population live.11
Albuquerque, New Mexico lies at this latitude, and is used as the
flight location for this work (35.0°N, 106.6°W).
2. Aircraft Dynamics
The flight dynamics model for the SRHALE system is adapted from the
six degree-of-freedom model developed by Beard and McLain.21 The
aerodynamic forces are computed as follows:
q = 1
Table 1: Parameters for the SRHALE aircraft.
Parameter Symbol Value
Battery specific energy Ubatt 350 W ·h kg
Maximum flight radius Rmax 3 km
Sweep angle Λ 20° Aspect ratio AR 30
Payload power Ppayload 250 W
Wing planform area A 60 m2
Solar panel area S 60 m2
Wingspan b 42 m
Chord c 1.4 m
Maximum battery energy Ebattmax 59.5 kWh
Minimum altitude hmin 60,000 ft
Maximum altitude hmax 90,000 ft
Minimum CL CLmin 0
Maximum CL CLmax 1.1
Minimum bank angle φmin -5° Maximum bank angle φmax 5 °
Minimum flight path angle γmin -5° Maximum flight path angle γmax
5°
where q is the dynamic pressure, ρ is the air density, V is the
aircraft velocity, and S is the surface area of the wing. The
variables L and D represent the lift and drag forces, while CL and
CD are the wing lift and drag coefficients, respectively.
The aircraft dynamics are modeled using a point-mass model which
describes the aircraft behavior in response to the inputs of
thrust, angle of attack, and bank angle. The equations of motion
for flight in the absence of wind are given as follows:
x = V cos ψ cos γ (4)
y = V sin ψ cos γ (5)
h = V sin γ (6)
V = Fthrust −D
ψ = L
mV cos φ− g
V cos γ (9)
where the states x, y, and h correspond to the aircraft north,
east, and altitude positions in the inertial reference frame. The
variable Fthrust represents thrust, φ is the aircraft bank angle, γ
is the flight path angle, ψ is the heading angle, m is the aircraft
mass, and g is the force of gravity.
The aircraft polars are determined using the approach described in
our previous work,20 however an older version of the aircraft model
is used in this paper. The base airfoil used is the SG6043, and
XFLR5 and XFOIL are used to obtain both the parasitic drag polar
and the lift curve. These data are then fit with nonlinear and
linear equations, respectively, and the drag coefficient CD is
computed as follows:
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CL = 0.729 + 5.44α (10)
CD = CDp + C2
π AR einv (12)
Here, CL is the lift coefficient, α is the angle of attack, CDp is
the parasitic drag coefficient, AR is the aspect ratio, and einv is
the inviscid span efficiency. The coefficients used in the
parasitic drag fit are listed in Table 2.
Table 2: Coefficient values for parasitic drag surface fit.
Coefficient Value
c1 9.68780E-02
c2 -1.19143E-02
c3 1.46589E-07
c4 -7.49336E-09
c5 -1.64442E-01
c6 3.97917E-05
c7 -4.18256E-06
3. Solar Flux
Solar flux values are computed using the Simple Model of the
Atmospheric Radiative Transfer of Sunshine (SMARTS).22 This model
is used by researchers in many different fields to predict
clear-sky spectral irradi- ance as a function of latitude,
longitude, altitude, time, and environmental factors. To increase
computation speed, solar flux values for the entire day were
precomputed at 1-minute intervals, and linear interpolation was
used to sample the data smoothly at smaller time intervals during
the dynamic path optimization.
In order to correct the raw solar flux values for the changing
aircraft orientation during flight, the obliquity factor is used as
described in our previous work.20 The equations are reproduced here
for completeness:
¯SN =
cos(−φ) cos(−θ)
µsolar = ¯SN
|| ¯SN || · N
||N || (15)
Here, ¯SN is the sun direction vector, which is calculated from the
solar azimuth (φs) and zenith (θs); N is the surface normal of the
wing, which is calculated using the aircraft orientation, including
the aircraft pitch angle, which is defined as θ = α+γ; and µsolar
is the obliquity factor, which is the product of the normalized sun
vector and the normalized surface normal.
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4. Air Properties
Estimates for air temperature and pressure are obtained from the
NASA 1976 Standard Atmosphere Model.23 The ideal gas law is used to
calculate air density: ρ = p/(R T ) where R is the specific gas
constant, and p and T are, pressure and temperature respectively,
at design altitude. Sutherland’s law is
used to calculate air viscosity: µair = µsl
( Tair
Tsl
)3/2 Tsl+Sc Tair+Sc where the subscript “sl” indicates values at
sea
level and Sc is the Sutherland temperature (110.4 Kelvin for
air).
5. Energy Balance
The system energy balance, as in our previous work,20 is computed
as follows:
Ptotal = Psolar + Pbattery − Ppropulsion − Ppayload (16)
where Psolar is the power entering the system through the solar
panels, Pbattery is the power leaving (+) or entering (−) the
battery, and Ppropulsion and Ppayload are the power requirements of
the propulsion system and payload, respectively. The balance of
power in the system, Ptotal, must satisfy
Ptotal ≥ 0 (17)
for feasible flight. This constraint is posed as an inequality to
give the optimizer more freedom, but the objective function
provides incentives to minimize this value, with the end result
that there is essentially no un-utilized power, and the constraint
is satisfied with equality.
Electrical energy storage in the battery is modeled as
follows:
Ebattery = Pbattery (18)
Due to the slow charge and discharge that occurs as the battery
cycles over an entire day, the battery is assumed to be perfectly
efficient, and no constraints are imposed on the charge and
discharge rates.
Potential energy storage is modeled with the following
relation:
Epotential = mg(h− h0) (19)
where h0 is the minimum height (60,000 ft). Combining the
electrical and potential energy storage gives the total
energy:
Etotal = Ebattery + Epotential (20)
This is used in the objective function for the dynamic path
optimization, as will be described later on.
B. Propeller Performance
1. Blade Element Momentum Method
We use CCBlade, an open source blade element momentum (BEM) code
formulated to give guaranteed convergence and in turn allow for a
continuously differentiable output.24a It includes a non-normal
inflow correction which allows us to mount the props in line with
the wing and include the angle of attack (AOA) of the wing as the
inflow angle to the prop. For atmospheric properties we use NASA’s
1976 Standard Atmosphere Modelb.25 This BEM formulation uses 2D
airfoil data to calculate the induction factors for each annular
disk of the propeller including Prandtl hub and tip loss correction
factors.26
aCCBlade.jl on BYU FLOW Lab GitHub
https://github.com/byuflowlab/CCBlade.jl bBYU FLOW Lab GitHub
Atmosphere.jl https://github.com/byuflowlab/Atmosphere.jl
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2. Airfoil Pre-computation
Blade element momentum theory is dependent on accurate airfoil data
for propeller performance predic- tion. To calculate the airfoil
data, we use XFOILc to run the Eppler 212 airfoil for 13 Reynolds
numbers ranging from 5 × 104 to 1 × 108, and fifty-one angles of
attack ranging between negative and positive stall (approximately
negative 10 to positive 20, depending on the Reynolds and Mach
number).
We use Airfoilpreppyd27 to model the stall delay experienced by
local sections on rotating blades. This code applies the rotational
correction on lift by Du et al.28 and drag by Eggers et al.29 as
well as extrapolation to high angles of attack by Viterna et
al.30
3. Propeller Performance Comparison
To validate that the BEM code with XFOIL airfoil data calculation
was consistent with Epema’s published experimental cases,31 we
compared his published experimental results with our computational
model of his setup, as described in our previous work.32 The
results for a constant RPM and varying freestream as seen in Fig.
1. Since the 2D airfoil data includes stall, the modeled effects of
stall can be seen on the 3D propeller performance in the very low
advance ratios. The more linear regions at the lowest advance
ratios are comprised of post-stall angles of attack calculated by
the Viterna extrapolation.
Airfoil Data
Momentum Balance
Pitch
Figure 1: Comparison of propeller efficiency with data collected by
Epema31 and the BEM code using XFOIL airfoil data. A maximum error
of 5% can be seen in the normal regions of operation which are
advance ratios below the region of steep drop-off.
4. Propeller Structures
We model propeller structures with a simple flap-wise cumulative
bending stress calculation and unidi- rectional carbon fiber
failure stress of 1.5× 109 Pa with a conservative safety factor of
10. The mass of the propellers is estimated as a solid propeller
with a density of 1440 kg/m3 for carbon fiber prepreg, similar to
propellers manufactured by Warp Drive Propellerse.
C. Power System
1. Solar Panel Model
In order to obtain the actual power output from the solar panels,
the solar panel efficiency is modeled following the methods
presented by Vika.33 This modeling framework relates solar panel
current and voltage as a function of solar flux and panel
temperature. This model is fit to data from Alta Devices 2017
Specifica- tions for single-junction GaAs cells, which represent a
light, flexible, efficient solar array appropriate for high
altitude aircraft applications. A constant atmospheric temperature
of -56.49 °C is assumed, as the altitudes in this work lie within
the tropopause.
cXfoil.jl on BYU FLOW Lab GitHub
https://github.com/byuflowlab/Xfoil.jl dAirfoilPreppy on BYU FLOW
Lab GitHub https://github.com/byuflowlab/AirfoilPreppy eWarp Drive
Propellers Inc. Commercial Website warpdriveinc.com, accessed
6/7/18
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2. Electric Motor
For calculating motor efficiency and power, we use a fundamental
first order motor model.34 Comparing the first order model to the
Maxon 305013 Brushless Motorf data, we found the efficiency,
current, and required voltage to all be within 1.5% for the nominal
RPM and torque.
In order to model the motor mass, we created a linear fit to the
motor data from the Astroflightg line of motors, as described in
our previous work.32 By way of review, Astroflight motors were
chosen due to the availability of data and the favorable range of
both power and Kv. We found a linear relationship between the mass
and the motor peak current divided by the motor Kv parameter. The
line fit in Eq. (21) shows the trend of the motor mass, with an R2
value of 0.94. The motors included in this empirically-based model
ranged from 1.5 kW to 15 kW and Kv from 32 to 1355.
mmotor = 2.464 Im Kv
+ 0.368 (21)
To accurately model motor performance in addition to mass, we
investigated the relationship between all of the motor parameters.
We found that there were no interdependencies other than those
first between the mass, motor peak current, and Kv, and second
between the no-load resistance and no-load current. The trend for
the latter is described by the fit in Eq. (22) with an R2 value of
0.93.
R0 = 0.0467(I0)−1.892 (22)
3. Linearized Battery and Motor Controller Masses
Due to the scope and nature of this comparative conceptual design
study, we used a simplified approach to model the motor controller
and battery masses. The motor controller model for mass was assumed
to be linear based on the specific power of 22,059 W/kg taken from
the Astroflight high voltage motor controller. Efficiency of the
motor controller was assumed to be a constant 97%. The battery was
modeled with a specific energy parameter of 300 Wh/kg,
representative of a mid-life, currently available Li-S
battery.35
III. Optimization Setup
A. Propulsion Optimization
The propulsion optimization is a static optimization, as opposed to
the trajectory which changes with time. Solving the propulsion
optimization repeatedly at every time step of the optimized path
presents a challenge in terms of computational power and time. In
order to feasibly solve the propulsion optimization in a reasonable
amount of time while using the dynamic trajectory for an entire day
as an input, the following simplification is made. Similar to
translating data into the frequency domain, the thrust and velocity
at each time for a given path are discretized into 100 bins, and
the weighting (or probability of occurrence) of each bin is
computed. This reduces the number of required propulsion system
evaluations from around 6,000 to just 100. We then neglect any bin
that has a probability of occurrence less than 0.35% (or about 5
minutes), which further reduces the multi-point evaluation to
around 20 points. This allows us to successfully optimize the
propulsion system at multiple points, considering the climbing,
descending, and night-time orbit flight modes, in an average time
of under one hour. Fig. 2 shows the thrust-velocity profile for a
given dynamic trajectory, with the bins and their probability of
occurrence designated. The white bins occur greater than 0.35% of
the time, and are the ones included in the optimization. The
night-time orbit is the dominant flight mode, occurring for more
than 40% of the day, with an average velocity and thrust of 30 m/s
and 80 N, respectively.
The propulsion optimization framework is formulated to minimize the
total energy expended during the day. The constraints are
formulated to meet the required thrust at each bin’s velocity, in
addition to a propeller material failure constraint and tip speed
constraint. Figure 3 gives a visual representation of the
optimization. The total energy is calculated using a weighted
average based on the probability of occurrence
fMaxon Motors Online Catalog http://www.maxonmotorusa.com, accessed
7/12/18 gAstroflight Motors http://www.astroflight.com, accessed
7/11/18
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0
50
100
150
200
250
300
)
Figure 2: Example thrust velocity input that has been discretized
into 100 bins. White boxes have a weighted occurrence, or
probability of greater than 5%, teal greater than 10%, and black
greater than 40%
of each bin and the required motor voltage and current. The mass of
the system is chosen at the maximum operating point that satisfies
the constraints. The propulsion optimization is performed in Julia
using the SNOPT solver.36
Energy Expended
Minimize:
System Weight Energy
Figure 3: Propulsion optimization framework with design variables
in red, models in green, and outputs in blue. For the multipoint
evaluation, the only duplicated design variable is RPM in order to
simulate a fixed pitch propeller.
To make the resulting optimized propulsion model simple enough for
the dynamic optimization, the non-dimensional propeller performance
curves were tabulated and passed into the dynamic optimization
framework. There, a continuous polynomial-spline is made and used
during the dynamic optimization (shown later in Fig. 12). The motor
performance model with splines satisfied the requirements of the
modeling language used for the dynamic path optimization, and
simply required the motor parameters to be passed in.
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B. Dynamic Path Optimization
1. Problem Formulation
The objective for SRHALE aircraft is to maximize endurance, subject
to the constraints of feasible flight operation, enabling the
aircraft to fly for long periods of time to accomplish a given
mission objective. In pursuit of this goal, dynamic optimization is
used to compute the optimal path for an aircraft to fly throughout
the whole day on the winter solstice, optimally angling toward the
sun to increase solar energy capture while flying as efficiently as
possible, and climbing to store potential energy when optimal. In
order to produce useful paths, the time-step of the dynamic
optimization must be small enough so that the discrete spatial
locations are close-enough to discern the aircraft behavior during
turns, and to reasonably satisfy the assumption of constant solar
flux and energy consumption between time-steps. In this work, the
time-step for the path optimization is 15 seconds. This fine time
resolution substantially increases the difficulty of solving the
optimization problem throughout the day. To overcome this
challenge, the problem is formulated using a receding horizon
approach,20 where the optimization problem is solved over a horizon
of 75 time-steps (or 18.75 minutes), data for the first 10 points
is saved, and then the horizon advances by 10 time-steps (or 2.5
minutes). This process is repeated until the path for the entire
day is computed.
2. Objective Function
Although the ideal objective for SRHALE flight in this case is to
maximize endurance, or the total flight time, with this formulation
that objective is not possible. Each horizon is independent from
each other, and coupling them over the entire day is
computationally infeasible. As a proxy, maximizing the total energy
in the system across all time serves to accomplish the same goal,
while allowing each horizon to be solved independently. Therefore,
the objective function for the dynamic optimization problem is
posed as follows:
maximize
∫ tf
t0
subject to √ x2 + y2 ≤ Rmax
hmin ≤ h ≤ hmax
CLmin ≤ CL(α) ≤ CLmax
φmin ≤ φ ≤ φmax
γmin ≤ γ ≤ γmax
(23)
A key difference between the objective described here and that in
our previous work20 is that the motor rotational speed, ωm, is
controlled instead of the thrust. This allows the removal of the
constraints on the maximum thrust, as these are naturally enforced
by the increase in energy required to achieve higher thrust (due to
the decrease in propulsion system efficiency at high advance
ratios, shown in Figure 12c which will be discussed later on in
greater detail).
3. Problem Solution
To formulate and solve the dynamic path optimization, the GEKKO
optimization suite is used.37,38
This is a freely-available Python software package for optimization
of systems with differential algebraic equations (DAE). The path
optimization problem is written in GEKKO, and the software uses
orthogonal collocation on finite elements to transcribe the DAE
problem into a purely algebraic system of equations, and automatic
differentiation to provide exact first and second derivatives. It
then uses a nonlinear solver to solve the system of equations. For
this work, the IPOPT solver is used, an open-source software
package for large-scale nonlinear optimization.39 The optimization
problem is solved in simultaneous mode, where the equations are
evaluated in parallel with the objective function
optimization.
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4. Initialization
For the first horizon, the nonlinear system is initialized with the
results of simulating the aircraft through- out the day in a
constant circular orbit, as outlined in our previous work.20 This
process is essential to suc- cessfully converging the system, and
the circular orbit also serves as a base-case to compare the
optimized trajectory against. This process is carried out using a
multi-layered approach where (1) a root-finder finds equilibrium
conditions for the three differential model terms by manipulating
φ, α, and in this work ωm
(instead of T ), and (2) the root-finder is wrapped in a
constrained minimization problem using Sequential Least Squares
Programming to determine the power-optimal steady-state trajectory
at the minimum alti- tude. This circular orbit is then (3)
integrated forward in time throughout the day using the solar data
from SMARTS to compute the total energy performance.
C. Battery Size Optimization
As part of the initialization for the trajectory optimization, the
battery size is optimized so that it fills completely during the
circular orbit without leaving any excess solar energy uncaptured.
This involves taking the same strategy for computing the
energy-optimal circular orbit and feeding that into an optimizer
that minimizes the squared difference between the maximum energy in
the battery and the total solar energy available for capture
throughout the flight, which is effectively equivalent to
maximizing the total energy in the battery at the end of the
day.
In our previous work, the battery-size optimization was carried out
to ensure that the base case was suf- ficiently well-designed to
give greater validity to subsequent improvements through trajectory
optimization. In this work, the battery size optimization serves as
the primary means through which the changes in the propulsion
system optimization and the subsequent adjustments by the
trajectory optimization translate into improved energy
performance.
D. System Optimization
As mentioned above, coupling the path and propulsion optimization
at the current level of model fidelity is not possible. A system
optimization that includes the effects of propulsion, trajectory,
and battery mass optimization therefore must adopt some form of
iterative scheme between the three sub-system optimizations.
To bridge the dynamic path and propulsion design optimization, a
point based iteration approach is first attempted. This involves
simply passing the designs/operating conditions back and forth
between the various models and recording the progression as seen in
Fig. 4a. The results of this approach are presented in the
following sections with surprisingly favorable trends.
Battery Sizing
Propulsion Optimization
Path Optimization
(a) Point Based Iteration
(b) System Level Optimization
Figure 4: Comparison of point based iteration and system
optimization framework.
IV. Results and Discussion
A. The Effect of Higher Fidelity Propulsion System Models on the
Dynamic Trajectory
This work builds upon the trajectory optimization approach in our
previous work,20 wherein we reported an end-of-day energy increase
of 8.2%. This increase describes the performance of an optimized
trajectory
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compared to a circular state machine strategy with optimal battery
mass. These results were obtained using propulsion models derived
from momentum-theory, which describe a variable-pitch propeller. A
key contribution of the current work is the integration of
higher-fidelity propulsion models including a fixed pitch
propulsion system into the trajectory optimization framework. This
makes possible the combination of design and trajectory
optimization through a sequential iterative approach.
With these higher-fidelity propulsion system models, the trajectory
optimization now requires propulsion system parameters. For the
initial case, these parameters were obtained by optimizing the
propulsion system for a circular orbit at the minimum flight
altitude. The parameters from this optimization are shown in Tables
3 and 4. The energy performance of the trajectory optimization is
shown in Figure 5. The complete cycle of propulsion system
optimization, battery mass optimization, and trajectory
optimization will be referred to as iteration 1 of the combined
sequential optimization.
Table 3: Initial motor parameters obtained from optimizing the
propulsion system for a circular path.
Parameter Value
Table 4: Initial propeller parameters obtained from optimizing the
propulsion system for a circular path.
r Chord Twist
0.200000 0.0390 66.9486
0.288875 0.2121 56.5429
0.377750 0.2511 49.0805
0.466625 0.2771 42.9949
0.555500 0.2617 38.3696
0.644375 0.2288 34.5409
0.733250 0.1862 31.6920
0.822125 0.0985 29.5858
0.911000 0.0870 27.2552
0
10
20
30
40
50
Figure 5: Energy performance of initial trajectory optimization,
iteration 1.
The end-of-day energy increase obtained by the trajectory
optimization in this initial case (iteration 1) is 6.9%. This is
slightly lower than the 8.2% we reported previously,20 and is
consistent with the fact that the propulsion system is now modeled
with a fixed-pitch propeller. Since the propulsion system was
optimized for
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a circular orbit, the climbing and descending from the trajectory
optimization is less efficient than with the momentum-theory
propulsion models that are more like a variable-pitch propeller. An
additional reason for this difference is due to slight differences
in the drag model used. For a point of reference, earlier
trajectory optimization results that were performed with the
momentum-theory propulsion model but using the exact same drag
model as the current paper showed an energy increase of 9.5%. This
suggests that the initial energy increase of 6.9% reported above
may be biased upward by perhaps 1.3%. Similarly to our previous
work,20 the trajectory optimization yields four distinct flight
modes. These flight modes are shown in Figure 6 for the first
iteration of the combined optimization.
E (km)
3 2 10 1 2 3 Al
t ( km
Start Finish
(a) Charge
E (km)
3 2 10 1 2 3
Al t (
Start Finish
(b) Climb
E (km)
3 2 10 1 2 3
Al t (
Start Finish
(c) Descent
E (km)
3 2 10 1 2 3
Al t (
Start Finish
(d) Night Power Conservation
Figure 6: Stages of the 24 hour optimized trajectory for the first
iteration.
The first flight mode is the charge stage (Figure 6a). This occurs
at dawn and continues until the battery is nearly full. The
aircraft follows a fabiform path while climbing and descending,
changing the angle of attack to increase the solar energy obtained.
A net maximum energy is achieved by balancing an increase in angle
of attack with the accompanying increase in induced drag losses.
This path precesses with the sun, and the maximum solar energy is
obtained while the aircraft is flying away from the sun with a high
angle of attack and lower airspeed. The path is wavy to maximize
the total time spent on this part. After turning around, the
aircraft flies towards the sun in a direct path with a low angle of
attack and higher airspeed, minimizing the amount of time spent in
an orientation with sub-optimal solar capture. This flight mode
lasts about 6 hours.
The second flight mode is the climb stage (Figure 6b). This stage
begins when the battery is nearly full, and the aircraft begins to
store potential energy by climbing in altitude aggressively. The
climb continues in
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a circular path that angles toward the sun and precesses with it,
and includes a climb and descent strategy throughout the climb. The
maximum thrust is reached during this time, and with it the
greatest requirements for the propulsion system. This flight mode
lasts about 2 hours.
The third flight mode is the descent stage (Figure 6c). This stage
begins once the solar energy available is not sufficient to keep
the battery fully charged. The aircraft begins to descend,
optimally balancing electrical and potential energy storage. It
follows a modified circular path that angles toward the sun to
collect as much solar energy as possible despite the low sun angle.
Once the sun has set completely, the aircraft transitions to a
perfect circular orbit in a zero-thrust glide downward to the
minimum altitude constraint. This flight mode lasts about 2
hours.
The final flight mode is the night time power conservation stage
(Figure 6d). This begins once the aircraft reaches the minimum
altitude constraint. It flies in a completely circular path, with
an optimal velocity and bank angle that minimizes the power needed.
This is the longest of the flight modes, and lasts about 14 hours.
It continues through the night until the sun comes out in the
morning, whereupon the battery begins to charge again.
These flight modes remain similar to those in our previous work,20
notwithstanding the higher-fidelity propulsion system models. One
interesting difference is that tight circular orbits are much more
prevalent just before sunset, as seen in the Descent stage (Figure
6c). This corresponds with the previously published observation
that near sunset, tight circles are able to point the solar array
in a way that gathers more energy than larger turns do.40
B. Combined Sequential Optimization
The energy performance of the combined optimization is shown in
Figure 7, which details the total energy and the battery
state-of-charge throughout the day for each iteration. The total
energy includes electrical energy stored in the battery and
potential energy stored by climbing above the minimum altitude of
60,000 ft. The energy performance increases with each iteration,
with the largest increase occurring between iteration 1 and 2 and
the second largest between iterations 6 and 7.
0 4 8 12 16 20 24 Time (hr)
10
0
10
20
30
40
50
h)
Iter 1 Iter 2 Iter 3 Iter 4 Iter 5 Iter 6 Iter 7
(a) Total energy
20
0
20
40
60
80
100
)
Iter 1 Iter 2 Iter 3 Iter 4 Iter 5 Iter 6 Iter 7
(b) Battery state-of-charge
Figure 7: Total energy and battery state-of-charge for each
combined iteration.
The maximum energy increases as the combined optimization
progresses (Figure 7a), primarily due to the increase in battery
mass, which is shown in Figure 8b. The battery mass is sized
according to the steady-state orbit, and as the propulsion
efficiency increases, the energy required for flight decreases,
which means that the aircraft can store additional solar energy in
the battery. As the aircraft mass increases, it requires more
energy to fly, and thus the battery mass addition requires finding
the optimum between the two.
Once the battery is full, the aircraft begins to climb, as shown in
Figure 8a. The maximum height between iterations decreases with the
first 3 iterations, and remains nearly constant from then on. One
reason for this decrease is because the increased battery mass
increases the amount of gravitational potential
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American Institute of Aeronautics and Astronautics
energy stored per unit height, decreasing the amount of height
required to store the same amount of potential energy. Other
factors at play include the increase in motor and propeller
efficiencies (summarized in Figure 11), which allow more efficient
potential energy storage by reducing the energy requirements for
the climb and descent (see Appendix, Figure 14c).
0 3 6 9 12 Time (hr)
60
62
64
66
68
70
72
74
ft)
Iter 1 Iter 2 Iter 3 Iter 4 Iter 5 Iter 6 Iter 7
(a) Height
127
128
129
130
131
132
133
134
Figure 8: Height and battery mass for each combined
iteration.
Figure 9 compares the performance of the steady-state and optimized
trajectories for each iteration. While the energy performance
increases with each iteration, the energy benefit from the
trajectory optimization remains nearly constant, decreasing
slightly with each iteration (Figure 9b). One explanation for this
decrease is that the propulsion system optimization increases the
energy efficiency of the night-time circular orbit, and the
trajectory optimization does not affect the night-time behavior.
Since the steady-state trajectory becomes more energy efficient,
there is less room for the optimized trajectory to improve upon
it.
1 2 3 4 5 6 7 Iteration
25
20
15
10
5
0
1 2 3 4 5 6 7 Iterations
0
1
2
3
4
5
6
7
8
(b) Energy benefit from trajectory optimization
Figure 9: Energy performance of each iteration of the combined
sequential optimization.
The energy benefit of the combined sequential optimization is
summarized in Figure 10. The total energy increase is 20.4%, which
is quite significant. This combined effect is more than 3 times the
benefit of optimizing the trajectory or the propulsion system
alone, and indicates the benefit that combining design and
trajectory optimization can have on HALE flight. With the benefits
of the combined optimization
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established, a closer look will be taken at where they are
manifest, and how the trajectory and propulsion system
optimizations interact through sequential iterations to produce the
benefit.
1 2 3 4 5 6 7 Iterations
0
5
10
15
20
16.51
20.42
Figure 10: Energy benefits of combined propulsion and trajectory
optimization. Each value is the difference in the end-of-day
battery state-of-charge between the optimized trajectory and the
initial steady-state orbit (iteration 1 in Figure 9a).
Figure 11 shows the progression of the point based iterations.
Three readily apparent main factors should be noted: first, the
battery mass significantly increases; second, the flight path
height decreases; third, the overall propulsion efficiency
increases.
Figure 11: Summary of results for each iteration. Percent energy
increase and increased flight time are in reference to the
optimized trajectory in the first iteration of the sequential
optimization scheme.
These three effects are inter-related at a system level: greater
battery mass requires the aircraft to fly faster with increased
thrust but also provides the capability to store more solar energy.
The increased storage will only be useful if the propulsion system
is able to increase efficiency to offset the extra power required
to fly a heavier aircraft. A main driver in the average efficiency
of a fixed pitch propeller is the breadth of the flight
requirements, which drives the aircraft to climb much more
conservatively.
Figure 12 shows heat maps of the flight requirements on the
propeller performance curves at each iteration. A greater thickness
or density of the line indicates more operation at that point. The
left-most dense region in the heat map is when the aircraft is
climbing, the center dense region is during the night time circular
orbit, and the right-most dense region is in descent. Increasing
the propeller average efficiency (see Fig. 12c) requires both a
very high curve as well as a very large weighted integral below the
curve. As seen in the final iteration, iteration 7, the propeller
peak efficiency is slightly higher than the rest, but the spread of
the curve is less than the previous iteration, iteration 6. Since
the weighted integral is the objective in order to minimize the
total power of the system, iteration 7 outperforms iteration 6 by
several percent.
V. Conclusion and Future Work
A framework for combined path and propulsion optimization has been
described, and the initial results show promise for increasing the
feasibility of solar-regenerative high-altitude long endurance
(SRHALE) aircraft in station-keeping missions. Compared to a
steady-state orbit with a propulsion system optimized for such,
after 7 iterations of the point-based combined optimization method
the end-of-day energy increased
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0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Advance Ratio
0.000
0.025
0.050
0.075
0.100
0.125
(a) Thrust Coefficient
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Advance Ratio
0.000
0.005
0.010
0.015
0.020
0.025
0.030
7Iteration:
0.0
0.2
0.4
0.6
0.8
1.0
Figure 12: Torque coefficient, thrust coefficient, and propeller
efficiency by iteration, with heat maps indi- cating frequency of
use.
by 20.2%, translating to an increase in flight time of 2.36 hours.
This is a significant improvement, and is evidence of the powerful
role of combining aircraft design and trajectory
optimization.
The current work is not without limitations, and efforts are
currently underway to address them. Since the changes in battery
mass are a key driver of the energy improvements, future work is
needed to ensure that the effects of changing mass on the
propulsion system optimization are fully accounted for. Changes in
the aircraft mass influence the thrust required to stay airborne,
and the feasibility of SRHALE flight is most sensitive to the mass.
Therefore, more work is needed to more clearly separate out the
effects of changes in the battery mass from the effects of changes
in the propulsion design and the optimal path. In future work,
additional consideration will be given to the treatment of the
battery mass generally and the optimization strategy behind it.
Additionally, the assumption that structural mass remains constant
while battery mass increases is likely optimistic, and a
commensurate structural mass increase of at least one-to-one may be
warranted.
The base-case design used for comparison deserves additional
consideration, as its level of optimality is a strong determiner of
the potential gains from combined optimization. Future work will
include a more detailed approach to establish the base-case design
of the propulsion system and battery size for a circular orbit.
This will give increased weight to the comparisons against the
base-case performance. Finally, future plans include extending this
combined optimization framework to a multidisciplinary system
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American Institute of Aeronautics and Astronautics
design architecture using a high-level system optimizer (such as an
Analytical Target Cascading algorithm41) in lieu of point-based
iteration to further extend the feasibility of SRHALE aircraft
through combined path and propulsion design optimization.
Funding Sources
The authors gratefully acknowledge support from the Facebook
Connectivity Lab. Any opinions, findings, or conclusions expressed
herein are those of the authors and do not necessarily reflect the
views of Facebook.
Acknowledgments
The authors also express gratitude to Abe Martin, Tim McLain, Randy
Beard, Taylor McDonnell, and Judd Mehr for their helpful feedback
and invaluable assistance in formulating the models used in this
paper.
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References
1Vuolo, F., Essl, L., and Atzberger, C., “Costs and benefits of
satellite-based tools for irrigation management,” Frontiers in
Environmental Science, Vol. 3, No. July, 2015, pp. 1–12.
2Damjanov, K., “Of Defunct Satellites and Other Space Debris,”
Science, Technology, & Human Values, Vol. 42, No. 1, 2017, pp.
166–185.
3Boucher, R., “History of solar flight,” 20th Joint Propulsion
Conference, 1984, pp. 1–22. 4D’Oliveira, F. A., De Melo, F. C. L.,
and Devezas, T. C., “High-altitude platforms – Present situation
and technology
trends,” Journal of Aerospace Technology and Management , Vol. 8,
No. 3, 2016, pp. 249–262. 5Noll, T. E., Brown, J. M., Perez-davis,
M. E., Ishmael, S. D., Tiffany, G. C., and Gaier, M.,
“Investigation of the
Helios Prototype Aircraft Mishap Volume I Mishap Report National
Oceanic and Atmospheric Administration and,” Tech. Rep. January,
NASA, 2004.
6Rapinett, A., Zephyr: a high altitude long endurance unmanned air
vehicle, Ph.D. thesis, Citeseer, 2009. 7Gomez, M. and Cox, A.,
“Flying Aquila: Early lessons from the first full-scale test flight
and the path ahead,” 2016. 8McKeegan, N., “Odysseus: Aurora’s
radical, unlimited endurance, solar powered aircraft, gizmag. com,
vol,” 2008. 9Haddox, C., “SolarEagle (Vulture II) Backgrounder,”
2010.
10Cesnik, C. E., Palacios, R., and Reichenbach, E. Y., “Reexamined
Structural Design Procedures for Very Flexible Aircraft,” Journal
of Aircraft , Vol. 51, No. 5‘, 2014, pp. 1580–1591.
11McDonnell, T. G., Mehr, J. A., and Ning, A., “Multidisciplinary
Design Optimization Analysis of Flexible Solar- Regenerative
High-Altitude Long-Endurance Aircraft,” 2018 AIAA/ASCE/AHS/ASC
Structures, Structural Dynamics, and Materials Conference, 2018, p.
0107.
12McDonnell, T., Mehr, J., and Ning, A., “Multidisciplinary Design
Optimization of Flexible Solar-Regenerative High- Altitude
Long-Endurance Aircraft,” Jan 2018.
13Hesse, H. and Palacios, R., “Reduced-Order Aeroelastic Models for
Dynamics of Maneuvering Flexible Aircraft,” AIAA Journal , Vol. 52,
No. 8, 2014, pp. 1717–1732.
14Caverly, R. J., Forbes, J. R., Danowsky, B. P., and Suh, P. M.,
“Gust-load alleviation of a flexible aircraft using a disturbance
observer,” AIAA Guidance, Navigation, and Control Conference, 2017
, 2017.
15Haghighat, S., T. Liu, H. H., and R. A. Martins, J. R.,
“Model-Predictive Gust Load Alleviation Controller for a Highly
Flexible Aircraft,” Journal of Guidance, Control, and Dynamics,
Vol. 35, No. 6, 10 2012, pp. 1751–1766.
16Brandt, S. A. and Gilliam, F. T., “Design analysis methodology
for solar-powered aircraft,” Journal of Aircraft , Vol. 32, No. 4,
7 1995, pp. 703–709.
17Baldock, N. and Mokhtarzadeh-Dehghan, M., “A study of
solar-powered, high-altitude unmanned aerial vehicles,” Aircraft
Engineering and Aerospace Technology, Vol. 78, No. 3, 2006, pp.
187–193.
18Morrisey, B. and Mcdonald, R., Multidisciplinary Design
Optimization of an Extreme Aspect Ratio HALE UAV , Ph.D. thesis,
2009.
19Ozoroski, T. A., Nickol, C. L., and Guynn, M. D., “High Altitude
Long Endurance UAV Analysis Model Development and Application Study
Comparing Solar Powered Airplane and Airship Station-Keeping
Capabilities,” , No. January, 2015.
20Martin, R. A., Gates, N. S., Ning, A., and Hedengren, J. D.,
“Dynamic Optimization of High-Altitude Long Endurance Aircraft
Trajectories Under Station Keeping Constraints,” Journal of
Guidance, Control, and Dynamics, 2018, forthcoming.
21Beard, R. W. and McLain, T. W., Small unmanned aircraft: Theory
and practice, Princeton university press, 2012. 22Gueymard, C. A.,
“Prediction and validation of cloudless shortwave solar spectra
incident on horizontal, tilted, or tracking
surfaces,” Solar Energy, Vol. 82, No. 3, 2008, pp. 260 – 271.
23“U.S. Standard Atmosphere, 1976,” Tech. rep., National Oceanic
and Atmospheric Administration; National Aeronautics
and Space Administration; United States Air Force, Washington D.C.,
1976. 24Ning, A., “A Simple Solution Method for the Blade Element
Momentum Equations with Guaranteed Convergence,” Wind
Energy, Vol. 17, No. 9, Sep 2014, pp. 1327–1345. 25“US standard
atmosphere 1976,” Tech. Rep. NASA-TM-X-74335, NASA, 1976.
26Glauert, H., “Airplane Propellers. In: Aerodynamic Theory,” Vol.
4, Springer, Berlin, Heidelberg, 1935, pp. 169–360. 27Ning, S. A.,
“AirfoilPrep.py Documentation: Release 0.1.0,” Tech. rep., sep
2013. 28Du, Z. and Selig, M., “A 3-D Stall-Delay Model for
Horizontal Axis Wind Turbine Performance Prediction,” 1998
ASME
Wind Energy Symposium, American Institute of Aeronautics and
Astronautics, Jan 1998. 29Eggers, A. J., Chaney, K., and
Digumarthi, R., “An Assessment of Approximate Modeling of
Aerodynamic Loads on the
UAE Rotor,” ASME 2003 Wind Energy Symposium, ASME, 2003. 30Viterna,
L. A. and Janetzke, D. C., “Theoretical and Experimental Power from
Large Horizontal-Axis Wind Turbines,”
Tech. rep., Sep 1982. 31Epema, H., Wing Optimisation for Tractor
Propeller Configurations, Master’s thesis, Delft University of
Technology,
Jun 2017. 32Moore, K. R. and Ning, A., “Distributed Electric
Propulsion Effects on Existing Aircraft Through
Multidisciplinary
Optimization,” 2018 AIAA/ASCE/AHS/ASC Structures, Structural
Dynamics, and Materials Conference, 2018, p. 1652. 33Vika, H. B.,
Modelling of Photovoltaic Modules with Battery Energy Storage in
Simulink / Matlab Havard Breisnes Vika,
Ph.D. thesis, Norwegian University of Science and Technology, 2014.
34Drela, M., First Order DC Electric Motor Model , MIT Aero and
Astro, Cambridge Massachusetts, Feb 2007. 35Mikhaylik, Y. V.,
Kovalev, I., Schock, R., Kumaresan, K., Xu, J., and Affinito, J.,
“High Energy Rechargeable Li-S Cells
for EV Application: Status, Remaining Problems and Solutions,” ECS,
2010. 36Gill, P. E., Murray, W., and Saunders, M. A., “SNOPT: An
SQP algorithm for largescale constrained optimization,”
SIAM Journal on Optimization, Vol. 12, No. 4, 2002, pp.
979–1006.
19 of 24
American Institute of Aeronautics and Astronautics
37Beal, L., Hill, D., Martin, R., and Hedengren, J., “GEKKO
Optimization Suite,” Processes, Vol. 6, No. 8, 2018, pp. 106.
38Hedengren, J. D., Shishavan, R. A., Powell, K. M., and Edgar, T.
F., “Nonlinear modeling, estimation and predictive
control in APMonitor,” Computers & Chemical Engineering, Vol.
70, 2014, pp. 133–148. 39Wachter, A. and Biegler, L. T., “On the
implementation of an interior-point filter line-search algorithm
for large-scale
nonlinear programming,” Mathematical programming, Vol. 106, No. 1,
2006, pp. 25–57. 40Edwards, D. J., Kahn, A. D., Kelly, M., Heinzen,
S., Scheiman, D. A., Jenkins, P. P., Walters, R., and Hoheisel,
R.,
“Maximizing Net Power in Circular Turns for Solar and Autonomous
Soaring Aircraft,” Journal of Aircraft , Vol. 53, No. 5, jan 2016,
pp. 1237–1247.
41Martins, J. R. R. A. and Lambe, A. B., “Multidisciplinary Design
Optimization: A Survey of Architectures,” AIAA Journal , Vol. 51,
No. 9, sep 2013, pp. 2049–2075.
20 of 24
Appendix
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-360 W -227 W
Iter 1 Iter 2 Iter 3 Iter 4 Iter 5 Iter 6 Iter 7
(a) Average power needed by stage
Morning Climb Descent Evening Stages
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-0.2 min -15.0 min
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(b) Total time in each stage
Morning Climb Descent Evening Stages
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30
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=-0.91 Wh
=-1.34 Wh
=-1.36 Wh
=-2.85 WhIter 1 Iter 2 Iter 3 Iter 4 Iter 5 Iter 6 Iter 7
(c) Total energy consumed by stage
Figure 13: Power consumption, with the difference () between the
first and last iteration shown.
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Morning Climb Descent Evening Stages
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8
ow er
G en
er at
io n
(k W
) =-93 W
=-757 W
=-184 W
=0 W
Iter 1 Iter 2 Iter 3 Iter 4 Iter 5 Iter 6 Iter 7
(a) Average solar power generated by stage
Morning Climb Descent Evening Stages
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-0.2 min -15.0 min
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-0.91 kWh +0.00 kWh
Iter 1 Iter 2 Iter 3 Iter 4 Iter 5 Iter 6 Iter 7
(c) Total solar energy generated by stage
Figure 14: Solar power generation, with the difference () between
the first and last iteration shown.
Consumed Generated Deficit 0
-6.46 kWh
-1.08 kWh
-5.37 kWh
Iter 1 Iter 2 Iter 3 Iter 4 Iter 5 Iter 6 Iter 7
Figure 15: Total energy
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Figure 16: Thrust-velocity profiles divided into bins, with a
cutoff probability of 0.5%: bins less than that are not included in
the optimization (and are grayed out). Colored boxes mark the bins
with the highest probabilities: black > 50%, red > 10%, cyan
> 5%.
23 of 24
30 35 40 45 50 Velocity (m/s)
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24 of 24
Brigham Young University
Nathaniel S. Gates
Kevin R. Moore
Original Publication Citation
BYU ScholarsArchive Citation
Aircraft Dynamics
Solar Flux
Air Properties
Energy Balance
Propeller Performance
Optimization Setup
Propulsion Optimization
Results and Discussion
The Effect of Higher Fidelity Propulsion System Models on the
Dynamic Trajectory
Combined Sequential Optimization